Towards the spectral properties and phase structure of QCD

In this thesis we explore a multitude of aspects concerning strongly coupled quantum field theories, with a special focus on QCD. The first part of the thesis is concerned with formal developments, with the noteworthy highlight of enabling the use of hydrodynamic numerical methods in Functional Renormalization Group equations. This lead to the subsequent discovery of discontinuous solutions for the effective potential in the vicinity of first order phase transitions